Changes

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ A, B, C \}.\!</math>

Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}.\!</math>

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

|

| align="center" |

<math>\begin{matrix}

<math>\begin{matrix}

\operatorname{e}

\operatorname{e}

& = & \operatorname{A}:\operatorname{A}

& = & \mathrm{A}\!:\!\mathrm{A}

& + & \operatorname{B}:\operatorname{B}

& + & \mathrm{B}\!:\!\mathrm{B}

& + & \operatorname{C}:\operatorname{C}

& + & \mathrm{C}\!:\!\mathrm{C}

\\[4pt]

\\[4pt]

\operatorname{f}

\operatorname{f}

& = & \operatorname{A}:\operatorname{C}

& = & \mathrm{A}\!:\!\mathrm{C}

& + & \operatorname{B}:\operatorname{A}

& + & \mathrm{B}\!:\!\mathrm{A}

& + & \operatorname{C}:\operatorname{B}

& + & \mathrm{C}\!:\!\mathrm{B}

\\[4pt]

\\[4pt]

\operatorname{g}

\operatorname{g}

& = & \operatorname{A}:\operatorname{B}

& = & \mathrm{A}\!:\!\mathrm{B}

& + & \operatorname{B}:\operatorname{C}

& + & \mathrm{B}\!:\!\mathrm{C}

& + & \operatorname{C}:\operatorname{A}

& + & \mathrm{C}\!:\!\mathrm{A}

\\[4pt]

\\[4pt]

\operatorname{h}

\operatorname{h}

& = & \operatorname{A}:\operatorname{A}

& = & \mathrm{A}\!:\!\mathrm{A}

& + & \operatorname{B}:\operatorname{C}

& + & \mathrm{B}\!:\!\mathrm{C}

& + & \operatorname{C}:\operatorname{B}

& + & \mathrm{C}\!:\!\mathrm{B}

\\[4pt]

\\[4pt]

\operatorname{i}

\operatorname{i}

& = & \operatorname{A}:\operatorname{C}

& = & \mathrm{A}\!:\!\mathrm{C}

& + & \operatorname{B}:\operatorname{B}

& + & \mathrm{B}\!:\!\mathrm{B}

& + & \operatorname{C}:\operatorname{A}

& + & \mathrm{C}\!:\!\mathrm{A}

\\[4pt]

\\[4pt]

\operatorname{j}

\operatorname{j}

& = & \operatorname{A}:\operatorname{B}

& = & \mathrm{A}\!:\!\mathrm{B}

& + & \operatorname{B}:\operatorname{A}

& + & \mathrm{B}\!:\!\mathrm{A}

& + & \operatorname{C}:\operatorname{C}

& + & \mathrm{C}\!:\!\mathrm{C}

\end{matrix}</math>

\end{matrix}</math>

|}

|}

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

|

| align="center" |

<math>\begin{bmatrix}

<math>\begin{bmatrix}

\mathrm{A}\!:\!\mathrm{A} &

\mathrm{A}\!:\!\mathrm{A} &